Since \(U\)  is not necessarily convex, we apply the new spike variation technique [1]}. Suppose that \(\bar{u}\in \mathcal {U}_{ad}\)  is an optimal control, for any \(\bar{t}\in [0,T]\) , define \(u^{\epsilon }\)  as follows. For \(s\in [0,T]\) ,
\(\begin{aligned}u^\epsilon _s={\left\lbrace \begin{array}{ll}u,&\text{if $(s,\omega )\in \mathcal {O}:=\rrbracket \bar{t},\bar{t}+\epsilon \rrbracket \backslash \sum _{n=1}^\infty \llbracket T_n\rrbracket $},\\\bar{u}_s,&\text{otherwise},\end{array}\right.}\end{aligned}\) 
